Trinidad and Tobago also won easily in the other Group C match, dispatching Guatemala 3-1.
Six-times Gold Cup winners Mexico dominated from the start, scoring four goals in the first half at a packed Soldier Field.
It was difficult to tell whether the result was proof of a Mexican resurgence, however, given that six members of the Cuban squad were unable to obtain visas to enter the United States, while another reportedly defected once the team arrived in Chicago.
The Cuban coach also could not get a visa.
Mexico wasted little time in putting Cuba to the sword, going 4-0 up in the first half with goals from Peralta (2), Carlos Vela and Andrés Guardado.
Peralta completed his hat-trick with a scrappy goal in the 61st minute, before Giovani Dos Santos capped the rout.
Mexico are likely to get a sterner test on July 15 when they meet Trinidad, who scored three goals inside the first 25 minutes against Guatemala.
Sheldon Bateau scored from a corner in the 11th minute before Cordell Cato doubled the lead three minutes later following horrendous defending by Guatemala.
Joevin Jones made it 3-0 shortly afterwards, before Carlos Ruiz pulled one back for Guatemala in the second half.
“From the start, we had to press high, press hard, stop Guatemala from getting into their rhythm and basically we compounded on their mistakes and got three goals pretty early,” said Trinidad striker Kenwyne Jones.
The Gold Cup, played every two years, is the regional championship for North and Central American and the Caribbean.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford)
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